Number Series

A number series question gives a list of numbers that follow a rule and asks for the next term (or a missing one). There is no formula to memorise, but there is a short list of patterns that cover almost every question you will meet. The skill is to try them in a sensible order and quickly.

A routine that works

1. Look at the size. Is the series rising slowly, rising fast or falling? Slow growth suggests addition; fast growth suggests multiplication or squares.
2. Write the differences between neighbouring terms.
3. If the differences are not constant, look at the differences of the differences.
4. Try ratios if the numbers grow quickly.
5. Check squares and cubes and numbers close to them.
6. Look for two interleaved series if the terms seem to jump around.

Pattern 1: arithmetic series

The same number is added each time. 4, 9, 14, 19, ? has a constant difference of 5, so the next term is 24.

Pattern 2: geometric series

Each term is multiplied by the same number. 2, 6, 18, 54, ? multiplies by 3, so the next term is 162.

Pattern 3: growing differences

When the differences themselves follow a pattern, extend that pattern. 1, 2, 4, 7, 11, ? has differences 1, 2, 3, 4, so the next difference is 5 and the next term is 16. Similarly 100, 95, 85, 70, ? has differences −5, −10, −15, so the next difference is −20 and the next term is 50.

Pattern 4: squares, cubes and related series

• Squares: 1, 4, 9, 16, 25, ... next is 36.
• Cubes: 1, 8, 27, 64, 125, ...
• Squares with an adjustment: 2, 5, 10, 17, 26 is n² + 1.
• Products of neighbours, n × (n + 1): 2, 6, 12, 20, 30, 42 ...

If the numbers look close to squares or cubes you already know, test that first.

Pattern 5: multiply and add

Each term is the previous one times a number, plus or minus a constant. 3, 7, 15, 31, ? uses "× 2 + 1", so the next term is 63. Notice that the differences (4, 8, 16) double, which is a clue for this type.

Pattern 6: Fibonacci-type series

Each term is the sum of the two before it. 2, 3, 5, 8, 13, ? gives 21. The classic series 0, 1, 1, 2, 3, 5, 8 works the same way.

Pattern 7: alternating or interleaved series

Two separate series are woven together. In 2, 5, 4, 10, 6, 15, 8, ? the odd positions go 2, 4, 6, 8 and the even positions go 5, 10, 15, 20. So the next term, at an even position, is 20. When a series seems to zig-zag, split it into odd and even positions and look at each half separately.

Pattern 8: primes and other special numbers

2, 3, 5, 7, 11, 13, ? is the list of prime numbers, so the next is 17. Also look for triangular numbers (1, 3, 6, 10, 15, 21: the differences go up by 1 each time) and powers of 2 (2, 4, 8, 16, 32).

Worked examples

Example 1. 5, 11, 23, 47, ? Differences are 6, 12, 24, which double. Or each term is × 2 + 1. The next term is 47 × 2 + 1 = 95.

Example 2. 1, 3, 6, 10, 15, ? The differences are 2, 3, 4, 5, so the next difference is 6 and the next term is 21.

Example 3. 7, 10, 8, 11, 9, 12, ? Odd positions: 7, 8, 9. Even positions: 10, 11, 12. The next term (position 7) is 10.

Common mistakes

• Stopping at the first pattern that fits two terms. A rule must fit every term.
• Looking only at differences when the series is multiplicative.
• Missing interleaved series because you only checked consecutive terms.
• Arithmetic slips when computing differences. Write them above the series.

Exam tips

1. Spend about 20 seconds on each series. If no pattern appears, move on and come back.
2. Practise recognising squares up to 20² and cubes up to 10³ instantly.
3. When the answer options are given, test each one against the pattern you found.

Now try the quiz and read each explanation to see the rule behind it.